For a number to be in the mandelbrot set, it means that it stays within the boundary of that circle thing, and it repeats in a sequence, where a number not in the mandelbrot set dosen't stay in the boundary and dosen't repeat it's sequence

For a number to be in the mandelbrot set, it means that it stays within the boundary of that circle thing, and it repeats in a sequence, where a number not in the mandelbrot set dosen't stay in the boundary and dosen't repeat it's sequence

You used the word "it" four times without saying what "it" refers to! (And with two different meanings!) You don't say what "that circle thing" is. You don't say what sequence you are talking about and you don't say what "stay in the boundary" means. What you should be saying is that a certain sequence remains bounded. And I can find no requirement that it "doesn't repeat its sequence". If a sequence eventually repeats, it certainly remains bounded.

from that formula

from what formula?

, Z1 = Z0^2 + Z0 (each time 0 is incremented by 1

Okay, this is the"formula" you referred to above and that gives the sequence you are referring to. But as given that implies that Z2= Z12+ Z1 which is incorrect. You want Zn+1= Zn2+ c for a fixed number c and Z0= c. When you ask "is -i in the Mandelbrot set" you are taking c= -i. Then [itex]Z_0= -i[/itex], [itex]Z_1= (-i)^2+ (-i)= -1- i[/itex], [itex]Z_2= (-1+1)^2+ (-i)= i[/itex], [itex]Z_3= (i)^2+ (-i)= -1-i[/itex] again!